Quantile-quantile plot

An alternative way to compare the frequency distribution of a sample to a theoretical distribution such as the normal distribution is by plotting the order statistics $x_{(1)}$, ..., $x_{(n)}$ against the corresponding $1/(n+1)$, ..., $n/(n+1)$ quantiles of the theoretical distribution. If the sample follows the theoretical distribution then the points will fall approximately along a straight line. This plot is particularly good for revealing discrepancies in the lower and upper tails of the distributions. It can also be used to compare two samples by plotting the two sets of order statistics against one another. Figure 2.4 shows that the July daily maximum temperatures at Uccle have shorter lower and upper tails than a normal distribution.

Figure: Normal-quantile plot of 20th century daily maximum temperatures recorded at Uccle, Belgium, in July.

It can also be useful to plot the empirical distribution function (e.d.f) and the theoretically-derived cumulative distribution function (c.d.f). The e.d.f (or ogive) is a bar plot of the accumulated frequencies in the histogram and the c.d.f is the integral of the density function - e.g. the staircase and smooth curve respectively shown in the lower panel of Fig. 2.1. These cumulative distribution functions give directly empirical probabilities $p$ as a function of quantile value $x_p$. Mathematical definitions of these quantities will be given later in Chapter 4.